Modelling shock in solid targets Goran Skoro (UKNF Collaboration, University of Sheffield) NuFact 06 UC Irvine, August 24-30, 2006

Thermal Stress ~  E  T/(1- ) strain  T=EDD/C p  - thermal expansion coefficient E – elastic modulus  T- temperature rise - Poisson’s ratio EDD – energy deposition density C p – specific heat  =f(T); E=f(T); =f(T); Cp=f(T) !!! Thermal stress as a function of temperature for different materials? Estimation of the thermal stress in materials

20 J/g corresponds to ~ 300 J/cc in Ta (and W) - energy density for 4-5 MW beam power, 6-10 GeV protons - High temperature target candidates: Ta, W, Nb(?), Mo(?)…

Assuming that the tensile strength (=f(T)) is a measure of material mechanical strength we can introduce ‘stress quality’ factor = thermal stress/tensile strength lower value of stress quality factor -> ‘better’ candidate for solid target W looks better than Ta NB. Inconclusive strength data for Molybdenum. Looks interesting in general (it is valuable alloying agent). Almost all ultra-high strength steels contain Mo in amounts from 0.25 to 8%. Graphite is ‘special’ (T2K)

LS-DYNA supported longitudinalradial Beam energy [GeV] Peak Stress [MPa] Von Mises Stress in 2cm diameter, 66cm long graphite target beam power: 4 MW, 50 Hz; energy deposition from MARS; 4x2ns bunches per pulse; 10  s macro-pulse length. Stress in graphite target Stress is not the main problem for graphite target!

2cm 20cm The target material exposed to the beam will be ~ 20cm long and ~2cm in diameter (in tantalum case). Energy density per pulse ~ 300 J/cc. Rotating toroidal ring (operating at ~2000 K); Individual bars... Cooling: radiation The target is bombarded at up 50 Hz by a proton beam consisting of ~1ns long bunches in a pulse of a few micro-s length. micro-pulse macro-pulse Beam: protons, 3 – 30 GeV High temperature candidates: TANTALUM, TUNGSTEN,... Simulations of the shock in the solid Neutrino Factory target ISS baseline (April 2006): 4 MW, 10 GeV, 50 Hz, 4 bunches per pulse, 2 ns rms.

Energy deposition in solid target Temperature rise in solid target High energy particle cascade calculations (MARS) Input for thermal stress calculations (LS-DYNA) Here: TANTALUM, Beam power = 5 MW, repetition rate = 50 Hz Simulations... as realistic as possible

LS-DYNA simulations Material model used in the analysis Temperature Dependent Bilinear Isotropic Model Uses 2 slopes (elastic, plastic) for representing of the stress-strain curve Inputs: density, Young's modulus, CTE, Poisson's ratio, yield stress,... “Theory” LS-DYNA input (estimate; especially for T> 1000K) strain stress [MPa] reliable data can be found for temperatures up to 1000K (but inconclusive); no data (practically) at high temperatures. Problems with material data:

LS-DYNA simulations (TANTALUM) micro-pulse macro-pulse radial characteristic time longitudinal characteristic time characteristic time (shock transit time) = characteristic length / speed of sound in material optimal macro-pulse length (t opt ) t opt factor of 2 difference in shock magnitude (let's say) from 10 to 30  s

“Proof”: T2K target results Slicing of the target does not help if shock transit time is bigger than macro-pulse length Graphite Bar Target : r=15mm, L=900mm (2 interaction length) –Energy deposit … Total: 58kJ/spill, Max:186J/g   T  200K MARS J/g Distribution of the energy deposit in the target (w/ 1 spill) cm macro-pulse length = 5  scharacteristic time (radial) = 10  s beam length = 90 cmlength = 1.5 cmlength = 0.3 cm

Peak stress as a function of beam energy (normalized on mean energy density) for fixed beam power (5 MW) and repetition rate (50 Hz) for 2 different macro-pulse lengths (3  s) and (10  s) micro-pulse macro-pulse LS-DYNA simulations (TANTALUM)

measure of 'concentration' of deposited energy Stress per deposited power - at the level of 250 MPa per MW - 'mean peak value' = averaged peak (von Mises) stress across the target LS-DYNA simulations (TANTALUM)

Experiment (Tantalum wire) The wire is 0.5 mm diameter, tantalum. Originally it protruded from the graphite top connection by 0.5 mm and ended up protruding 3 mm. The wire ran for 16 hours at Hz repetition rate. The wire was run at 100 C rise per pulse for the first 6.5 hours,... The last 5 hours was at 4900 A, pulse, corresponding to a temperature rise or 150 C per pulse. The peak temperature... was estimated to be ~1300 C. One can see that the wire has become reduced in radius in parts and is thicker in others. Status of simulations of the current pulse – wire tests at RAL

LS-DYNA simulations Loads Current pulse: ~ 5 kA, exponential rise strain Time, 100 ns intervals Rise time: ~100 ns Flat Top: ~500 ns 30 ns risetime fitted to the waveform Energy density;temperature rise across the wire Lorentz force induced pressure wave Geometry 0.5mm diameter; 40mm long wire; supported at bottom, free at top

LS-DYNA simulations Multiple pulses Pulse time (heating) ~ 600 ns; temperature rise per pulse ~ 110 C strain T=300K Time between pulses (cooling) ~ 300 ms; LS-DYNA needs 115 h to complete 1 pulse! APPROXIMATION: Time between pulses (cooling)~ 500  s; 50x longer than (longitudinal) characteristic time! 50 pulses (16 h to complete); temperature rise ~ 1300 C final cooling ↔ 500x longer time than (longitudinal) characteristic time.

LS-DYNA supported elements at the centreline 50 pulses plastic deformation free Tantalum wire: 0.5 mm diameter, 40 mm long Results EXPERIMENT: Originally it protruded from the graphite top connection by 0.5 mm and ended up protruding 3 mm. Simulation: 0.5 mm

supported elements at the surface 50 pulses free plastic deformationthicker in radius reduced LS-DYNA Tantalum wire: 0.5 mm diameter, 40 mm long Results EXPERIMENT: One can see that the wire has become reduced in radius in parts and is thicker in others. SIMULATION

Loads Current pulse: ~7 kA, 800 ns long Geometry 0.5mm diameter; 30mm long wire Initial temperature = 2300 K Tungsten wire test Peak current can be ‘tuned’ to have a wanted value of the peak stress in the wire

LS-DYNA supported Stress in 2 x 17 cm tungsten target (4 MW, 50 Hz, 6 GeV) Macro pulse length [  s] Peak Von Mises Stress [MPa] Comparison: Stress in real target vs. stress in tungsten wire Stress in tungsten wire (7.5 kA, 800 ns long pulse) Test result (7.5 kA, 800 ns long pulse) Tungsten wire (after a few minutes at 7.2 kA)

supported S. Brooks Stress in tungsten wire (5 kA, 800 ns long pulse) Stress in 3 x 20 cm tungsten target (4 MW) “Stress in 2 x 17 cm tungsten target” (2 MW) This wire had survived over 3 million pulses at 5 kA. Bent into this severe shape within a few minutes at 7.2 kA. Solution: bigger target radius? It looks possible. Captured yield practically the same for 1 cm radius -> 1.5 cm radius. Comparison: Stress in real target vs. stress in tungsten wire

LS-DYNA (3D) “Additional” stress in the target: for example when beam is not at a target axis (target bending, etc…) TUNGSTEN target operating at 2000 K 4x2ns long bunches in a 10  s long pulse Power = 4 MW, repetition rate = 50 Hz, Beam energy = 6 GeV (parabolic distribution) Energy deposition from MARS (S. Brooks) Peak Von Mises Stress [MPa] 2 x 17 cm Tungsten target Beam radius = Rod radius = 1 cm 3 x 20 cm Tungsten target Beam radius = Rod radius = 1.5 cm Beam offset [in rod radius units] 33 MPa; ~10% rise46 MPa; ~33% rise

LS-DYNA (3D) Beam radius = rod radius; Beam offset = 1/2 radius. Peak Stress = 182 MPaPeak Stress = 170 MPa Peak Stress = 175 MPa beam Tungsten target: 3x20 cm Beam: 4 MW, 50 Hz, 6 GeV, 4x2ns, 10  s (1) (2)(3) Energy deposition from MARS (S. Brooks) (1)(2)(3)

Comparison with the tests at the ISOLDE 0.9 Tantalum Cylinder, 1x10 cm EXPERIMENT Goran SKORO, Sheffield University LS-DYNA simulations surface displacement [  m]

Summary Solid target for the Neutrino Factory: Shock waves in candidate materials (Ta, W, C) characterised within limitations of material knowledge Effects of beam pulse length and multiple bunches/pulse understood (stress reduction by choosing optimal macro-pulse length) Test of wire: First estimate of the lifetime of tantalum (and tungsten) NF target VISAR is purchased to measure surface velocity of wire and compare results with LS-DYNA calculations (this will help to extract high temperature material data from experiment) Repeat experiment with graphite and the others candidate materials Conclusions Nice agreement between LS-DYNA and existing experimental results 2 MW -> looks possible in 2 cm diameter target (W is better than Ta) 4 MW -> needs bigger target diameter (2 cm -> 3 cm)